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  • Kardar-Parisi-Zhang Universality

    I. Corwin

    Universality in Random Systems Universality in complex random systems is a striking concept which has played a central role in the direction of research within probability, mathematical physics and statistical mechanics. In this article we will describe how a variety of physical systems and mathematical models, including randomly growing interfaces, certain stochas- tic PDEs, traffic models, paths in random environments, and random matrices all demonstrate the same universal statistical behaviors in their long-time/large-scale limit. These systems are said to lie in the Kardar-Parisi-Zhang (KPZ) universality class. Proof of universality within these classes of systems (except for random matrices) has re- mained mostly elusive. Extensive computer simulations, nonrigorous physical arguments/heuristics, some labora- tory experiments, and limited mathematically rigorous results provide important evidence for this belief.

    Ivan Corwin is professor of mathematics at Columbia University, and research fellow of the Clay Mathematics Institute, Packard Foundation, as well as previous holder of the PoincarΓ© Chair at the Institut Henri PoincarΓ© and of the Schramm Fellowship at Mi- crosoft Research and MIT. His email address is ivan.corwin@gmail.com.

    For permission to reprint this article, please contact: reprint- permission@ams.org. DOI: http://dx.doi.org/10.1090/noti1334

    The last fifteen years have seen a number of break- throughs in the discovery and analysis of a handful of special integrable probability systems which, due to enhanced algebraic structure, admit many exact com- putations and ultimately asymptotic analysis revealing the purportedly universal properties of the KPZ class. The structures present in these systems generally origi- nate in representation theory (e.g. symmetric functions), quantum integrable systems (e.g. Bethe ansatz), alge- braic combinatorics (e.g. RSK correspondence), and the techniques in their asymptotic analysis generally involve Laplace’s method, Fredholm determinants, or Riemann- Hilbert problem asymptotics.

    This article will focus on the phenomena associated with the KPZ universality class [4] and highlight how certain integrable examples expand the scopeof and refine the notion of universality. We start by providing a brief introduction to the Gaussian universality class and the integrable probabilistic example of random coin flipping and the random deposition model. A small perturbation to the random deposition model leads us to the ballistic deposition model and the KPZ universality class. The ballistic deposition model fails to be integrable; thus to gain an understanding of its long-time behavior and that of the entire KPZ class, we turn to the corner growth model. The rest of the article focuses on various sides of this rich model: its role as a random growth process,

    230 Notices of the AMS Volume 63, Number 3

  • its relation to the KPZ stochastic PDE, its interpretation in terms of interacting particle systems, and its relation to optimization problems involving paths in random environments. Along the way, we include some other generalizationsof this processwhose integrability springs from the same sources. We close the article by reflecting upon some open problems.

    A survey of the KPZ universality class and all of the associated phenomena and methods developed or utilized in its study is far too vast to be provided here. This article presents only one of many stories and perspectives regarding this rich area of study. To even provide a representative cross-section of references is beyond this scope. Additionally, though we will discuss integrable examples, we will not describe the algebraic structures and methods of asymptotic analysis behind them (despite their obvious importance and interest). Some recent references which review some of these structures include [2], [4], [8] and references therein. On the more physics oriented side, the collection of reviews and books [1], [3], [5], [6], [7], [8], [9], [10] provides some idea of the scope of the study of the KPZ universality class and the diverse areas upon which it touches.

    We start now by providing an overview of the general notion of universality in the context of the simplest and historically first exampleβ€”fair coin flipping and the Gaussian universality class.

    Gaussian Universality Class Flip a fair coin 𝑁 times. Each string of outcomes (e.g. head, tail, tail, tail, head) has equal probability 2βˆ’π‘. Call 𝐻 the (random) number of heads and let β„™ denote the probability distribution for this sequence of coin flips. Counting shows that

    β„™(𝐻 = 𝑛) = 2βˆ’π‘(𝑁𝑛).

    Since each flip is independent, the expected number of heads is𝑁/2. Bernoulli (1713) proved that𝐻/𝑁 converges to 1/2 as 𝑁 goes to infinity. This was the first example of a law of large numbers. Of course, this does not mean that if you flip the coin 1,000 times, you will see exactly 500 heads. Indeed, in𝑁 coin flips one expects the number of heads to vary randomly around the value 𝑁/2 in the scale βˆšπ‘. Moreover, for all π‘₯ ∈ ℝ,

    lim π‘β†’βˆž

    β„™(𝐻 < 12𝑁+ 12βˆšπ‘π‘₯) = π‘₯

    ∫ βˆ’βˆž

    π‘’βˆ’π‘¦2/2

    √2πœ‹ 𝑑𝑦.

    DeMoivre (1738), Gauss (1809), Adrain (1809), andLaplace (1812) all participated in the proof of this result. The lim- iting distribution is known as the Gaussian (or sometimes normal or bell curve) distribution.

    A proof of this follows from asymptotics of 𝑛!, as derived by de Moivre (1721) and named after Stirling (1729). Write

    𝑛!= 𝛀(𝑛+ 1) = ∞

    ∫ 0

    π‘’βˆ’π‘‘π‘‘π‘›π‘‘π‘‘ = 𝑛𝑛+1 ∞

    ∫ 0

    𝑒𝑛𝑓(𝑧)𝑑𝑧

    where 𝑓(𝑧) = log𝑧 βˆ’ 𝑧 and the last equality is from the change of variables 𝑑 = 𝑛𝑧. The integral is dominated, as 𝑛 grows, by the maximal value of 𝑓(𝑧) on the interval [0,∞). This occurs at𝑧 = 1, thus expanding 𝑓(𝑧)β‰ˆβˆ’1βˆ’ (π‘§βˆ’1)22 , and plugging this into the integral yields the final expansion

    𝑛!β‰ˆ 𝑛𝑛+1π‘’βˆ’π‘›βˆš2πœ‹/𝑛. This general route of writing exact formulas for prob-

    abilities in terms of integrals and then performing asymptotics is quite common to the analysis of inte- grable models in the KPZ universality class, though those formulas and analyses are considerably more involved.

    The universality of the Gaussian distribution was not broadly demonstrated until work of Chebyshev, Markov, and Lyapunov around 1900. The central limit theorem (CLT) showed that the exact nature of coin flipping is immaterialβ€”any sum of independent identically dis- tributed (iid) random variables with finite mean and variance will demonstrate the same limiting behavior.

    Theorem 1. Let 𝑋1,𝑋2,… be iid random variables of finite mean π‘š and variance 𝑣. Then for all π‘₯ ∈ ℝ,

    lim π‘β†’βˆž

    β„™( 𝑁

    βˆ‘ 𝑖=1

    𝑋𝑖 < π‘šπ‘+π‘£βˆšπ‘π‘₯) = π‘₯

    ∫ βˆ’βˆž

    π‘’βˆ’π‘¦2/2

    √2πœ‹ 𝑑𝑦.

    Proofs of this result use different tools than the exact analysis of coin flipping, and much of probability theory deals with the study of Gaussian processes which arise through various generalizations of the CLT. The Gaussian distribution is ubiquitous, and, as it is the basis for much of classical statistics and thermodynamics, it has had immense societal impact.

    Random versus Ballistic Deposition The random deposition model is one of the simplest (and least realistic) models for a randomly growing one- dimensional interface. Unit blocks fall independently and in parallel from the sky above each site of β„€ according to exponentially distributed waiting times (see Figure 1). Recall that a random variable 𝑋 has exponential distribu- tion of rate πœ† > 0 (or mean 1/πœ†) if β„™(𝑋 > π‘₯) = π‘’βˆ’πœ†π‘₯. Such random variables are characterized by the memoryless propertyβ€”conditioned on the event that 𝑋 > π‘₯,π‘‹βˆ’π‘₯ still has the exponential distribution of the same rate. Con- sequently, the random deposition model is Markovβ€”its future evolution depends only on the present state (and not on its history).

    The random deposition model is quite simple to ana- lyze, since each column grows independently. Let β„Ž(𝑑, π‘₯) record the height above site π‘₯ at time 𝑑 and assume β„Ž(0, π‘₯) ≑ 0. Define random waiting times 𝑀π‘₯,𝑖 to be the time for the 𝑖-th block in column π‘₯ to fall. For any 𝑛, the event β„Ž(𝑑, π‘₯) < 𝑛 is equivalent to βˆ‘π‘›π‘–=1 𝑀π‘₯,𝑖 > 𝑑. Since the 𝑀π‘₯,𝑖 are iid, the law of large numbers and central limit theory apply here. Assuming πœ† = 1,

    lim π‘‘β†’βˆž

    β„Ž(𝑑, π‘₯) 𝑑 = 1 and limπ‘‘β†’βˆž

    β„Ž(𝑑, π‘₯) βˆ’ 𝑑 𝑑1/2 β‡’ 𝑁(π‘₯)

    jointly over π‘₯ ∈ β„€, where {𝑁(π‘₯)}π‘₯βˆˆβ„€ is a collection of iid standard Gaussian random variables. The top of Figure 2

    March 2016 Notices of the AMS 231

  • (a) (b)

    (c) (d)

    Figure 1. (A) and (B) illustrate the random deposition model, and (C) and (D) illustrate the ballistic deposition model. In both cases, blocks fall from above each site with independent exponentially distributed waiting times. In the first model, they land at the top of each column, whereas in the second model they stick to the first edge to which they become incident.

    shows a simulation of the random deposition model. The linear growth speed and lack of spatial correlation are quite evident. The fluctuations of this model are said to be in the Gaussian universality class since they grow like